You probably know many of the trigonometry functions for the more common angles. Some favorites are:
cos 0° = 1, and tan 45° = 1. That’s three values for three different angles in three different functions. This doesn’t even scratch the surface. Trig functions produce numerical values for all angles, whether in degrees or radians. When you want the secant of an offbeat angle, such as 47 degrees, you need to resort to some resource such as a table of values or a scientific calculator.
Where did all these function values come from? Who found them? Are there different formulas to use to find these values? Can they be computed by hand?
Going back to the basics of trigonometry, you find a great deal of information from a right triangle. The trig functions can be defined with the measures of the three sides of a right triangle. If you know the measure of an acute angle in a right triangle and you have the measures of the sides, then you can compute the function values with some careful dividing. That’s fine and dandy when you have a 30-60-90 right triangle or a 45-45-90 right triangle. The measures of the sides are
and
(or some multiple of these), respectively. Again, what about that 47-degree angle? If you very carefully construct a right triangle with an angle of 47 degrees, you may be able to get some fairly accurate measurements of the sides and see that this 47-43-90 right triangle has sides measuring about 0.7314-0.6820-1 (or some multiple thereof). How good are you at doing this? Do you trust your calculations?
Actually, trig function values have been around for centuries. In the mid-1500s, the mathematician François Viète created tables of all six functions correct to the nearest minute. Not familiar with the minute measure in trigonometry? Minutes and seconds are used to express fractions of degrees. One degree is equal to 60 minutes and one minute is equal to 60 seconds. So, to express an angle measuring
degrees, you write:
(seven degrees, 30 minutes, 30 seconds). Many cultures and mathematicians dabbled in creating tables of values for the trig functions. You can find tables that give function values to the nearest degree and the nearest minute. Many of the tables have three functions at the head of each column and the respective co-functions of the three functions at the bottom of the same columns. Functions and their co-functions have the same function values, just for different angle measures. This top-and-bottom arrangement saves space in a book.
Before handheld calculators, the trig tables found in textbooks or mathematical table publications were the only way to go for the layman. Calculators have made life so much easier. But calculators take some careful attention to details. The first detail, of course, is to have working batteries available. The next detail is to have the calculator set in the correct mode — degrees or radians — depending on what your application is and what form the angles that you’re using are in. When using the calculator to evaluate inverse trig functions, you really need to know about the domain and range of the function to get an accurate reading. The calculator will only tell you what you ask, not necessarily what you need to know.
Whichever method you use, just be sure to carefully follow any directions and have a working knowledge of the relationships between the different functions.